Revision as of 12:57, 21 November 2012

Bayesian model of univariate linear regression for QTL detection

See Servin & Stephens (PLoS Genetics, 2007).

Data: let's assume that we obtained data from N individuals. We note the (quantitative) phenotypes (e.g. expression level at a given gene), and the genotypes at a given SNP (as allele dose, 0, 1 or 2).

Goal: we want to assess the evidence in the data for an effect of the genotype on the phenotype.

Assumptions: the relationship between genotype and phenotype is linear; the individuals are not genetically related; there is no hidden confounding factors in the phenotypes.

Likelihood:

where β1 is in fact the additive effect of the SNP, noted a from now on, and β2 is the dominance effect of the SNP, d = ak.

Let's now write in matrix notation:

which gives the following conditional distribution for the phenotypes:

The likelihood of the parameters given the data is therefore:

Priors: we use the usual conjugate prior

Joint posterior:

Conditional posterior of B:

Here and in the following, we neglect all constants (e.g. normalization constant, YTY, etc):

We use the prior and likelihood and keep only the terms in B:

We expand:

We factorize some terms:

Let's define . We can see that ΩT = Ω, which means that Ω is a symmetric matrix.
This is particularly useful here because we can use the following equality: Ω − 1ΩT = I.

This now becomes easy to factorizes totally:

We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as:

Posterior of τ:

Similarly to the equations above:

But now, to handle the second term, we need to integrate over B, thus effectively taking into account the uncertainty in B: